A triple \((X, \mathsf {d}, \mathfrak {m})\)  is said to be a metric measure space if \((X, \mathsf {d})\)  is a complete separable metric space and \(\mathfrak {m}\)  is a Borel measure on \(X\)  with full support. The pioneer papers [1]}, [2]}, [3]} define \(\mathrm {CD}(K, N)\)  conditions for metric measure spaces from the point of view of optimal transportation theory (see also [4]}). Roughly speaking, a metric measure space \((X, \mathsf {d}, \mathfrak {m})\)  is said to be a \(\mathrm {CD}(K, N)\)  space if the Ricci curvature is bounded below by \(K \in \mathbb {R}\) , and the dimension is bounded above by \(N \in [1, \infty ]\) .
After that, adding Riemannian structure to \(\mathrm {CD}(K, N)\)  spaces, the definition of \(\operatorname{RCD}(K, N)\)  spaces is introduced in [5]}, [6]}, [7]}. Let us give an equivalent definition of \(\operatorname{RCD}(K, N)\)  spaces under assuming a bit of knowledges on the Sobolev space \(H^{1, 2}(X, \mathsf {d}, \mathfrak {m})\) .
